On the Structure of Commutative Banach Algebras Generated by Toeplitz Operators on the Unit Ball. Quasi-Elliptic Case. II: Gelfand Theory

Abstract

Extending our results in Bauer and Vasilevski (J Funct Anal 265(11):2956–2990, 2013) the present paper gives a detailed structural analysis of a class of commutative Banach algebras \(\mathcal {B}_k(h)\) generated by Toeplitz operators on the standard weighted Bergman spaces \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) over the complex unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). In the most general situation we explicitly determine the set of maximal ideals of \(\mathcal {B}_k(h)\) and we describe the Gelfand transform on a dense subalgebra. As an application to the spectral theory we prove the inverse closedness of algebras \(\mathcal {B}_k(h)\) in the full algebra of bounded operators on \(\mathcal {A}_{\lambda }^2(\mathbb {B}^n)\) for certain choices of \(h\). Moreover, it is remarked that \(\mathcal {B}_k(h)\) is not semi-simple. In the case of \(k=(n)\) we explicitly describe the radical \(\hbox {Rad}\, \mathcal {B}_n(h)\) of the algebra \(\mathcal {B}_n(h)\). This result generalizes and simplifies the characterization of \(\hbox {Rad}\,\mathcal {B}_2(1)\), which was given in Bauer and Vasilevski (Integr Equ Oper Theory 74:199–231, 2012).